 1028 R. F. WILLIAMS
 2D CONTINUED FRACTIONS AND POSITIVE
MATRICES
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Feb 1, 10

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Abstract. The driving force of this paper is a local symmetry in lattices. The
goal is two theorems: a partial converse to the PerronFrobenius the
orem in dimension 3 and a characterization of conjugacy in Sl3(Z). In
the process we develop a geometric approach to higher dimension con
tinued fractions, HDCF. HDCF is an active area with a long history:
see for example Lagarias, [L],[Br].
The algorithm: Let Zr be the set of all lattice points within r > 0
of a ray L = {mP 2 Rn : m > 0} Let z1 denote the point in Zr closest
to the origin. Having defined z1, .., zi, 1 i < n, let zi+1 be the point
of Zr closest point into the origin, which is independent of z1, ...zi.
Conjecture 1. z1, ..., zn is a basis of Zn.
The proof of this conjecture in dimension n = 3 occupies the bulk
of the paper. Here P 2 Rn is taken to have all positive components as
usual, and we use a metric in L?. For further discussion and the relation
of the conjecture to the Minkowski sequential minima theorem, see the
section Remarks on the algorithm below.
We have no such arithmetic theorems in dimension n > 3. However,
in the first place, the symmetry theorem, its relation to arithmetic, and
part of the bifurcation theorem are proved in all dimensions. Secondly,
lots of computer studies in the next three dimensions have been done
and we find no obstruction to this conjecture; in particular, for the
partial converse mentioned above, we easily find bases of Zn, relative
to which the matrix becomes positive, for n = 3, 4, 5, 6. A few examples
are given below.
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